Question: Solve for $x$ : $ 8|x + 5| - 2 = 2|x + 5| + 8 $
Solution: Subtract $ {2|x + 5|} $ from both sides: $ \begin{eqnarray} 8|x + 5| - 2 &=& 2|x + 5| + 8 \\ \\ { - 2|x + 5|} && { - 2|x + 5|} \\ \\ 6|x + 5| - 2 &=& 8 \end{eqnarray} $ Add ${2}$ to both sides: $ \begin{eqnarray} 6|x + 5| - 2 &=& 8 \\ \\ { + 2} &=& { + 2} \\ \\ 6|x + 5| &=& 10 \end{eqnarray} $ Divide both sides by ${6}$ $ \dfrac{6|x + 5|} {{6}} = \dfrac{10} {{6}} $ Simplify: $ |x + 5| = \dfrac{5}{3}$ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ x + 5 = -\dfrac{5}{3} $ or $ x + 5 = \dfrac{5}{3} $ Solve for the solution where $x + 5$ is negative: $ x + 5 = -\dfrac{5}{3} $ Subtract ${5}$ from both sides: $ \begin{eqnarray} x + 5 &=& -\dfrac{5}{3} \\ \\ {- 5} && {- 5} \\ \\ x &=& -\dfrac{5}{3} - 5 \end{eqnarray} $ Change the ${ - 5}$ to an equivalent fraction with a denominator of $3$ $ x = - \dfrac{5}{3} {- \dfrac{15}{3}} $ $ x = -\dfrac{20}{3} $ Then calculate the solution where $x + 5$ is positive: $ x + 5 = \dfrac{5}{3} $ Subtract ${5}$ from both sides: $ \begin{eqnarray} x + 5 &=& \dfrac{5}{3} \\ \\ {- 5} && {- 5} \\ \\ x &=& \dfrac{5}{3} - 5 \end{eqnarray} $ Change the ${ - 5}$ to an equivalent fraction with a denominator of $3$ $ x = \dfrac{5}{3} {- \dfrac{15}{3}} $ $ x = -\dfrac{10}{3} $ Thus, the correct answer is $x = -\dfrac{20}{3} $ or $x = -\dfrac{10}{3} $.